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Proof step 8 from proof modmod-add2-negpos-pos-01 (previous step)

i < 0, i + j0 < 0, j0 < j, 0 < i1 * j + j0, 0 < j, 0 < i1, j ≤ i1 * j + j0, 0 ≤ i1 * j + i + j0, 0 ≤ j0, 0 ≤ i1 ⊦ (i + j0) % j = 0, (i1 * j + i + j0) % j = (i + j0) % j + j, j = 1

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: mod-add1-posneg-pos seq: ⊦ i % j = 0 ∧ 0 ≤ i + i0 ∧ 0 ≤ i ∧ i0 < 0 ∧ i0 % j ≠ 0 ∧ 0 < j → (i + i0) % j = i0 % j + j subst: [j, i0, i] → [j, i + j0, i1 * j] flags: <all paths>

Additional information

Used simplifier rules: mod-mult-04: j ≠ 0 ⊦ (i * j) % j = 0;% ⊦ i * 0 = 0; (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult% ⊦ i + 0 = i; (s)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s)% ⊦ 0 < i → (0 ≤ i * j ↔ 0 ≤ j); (s)% ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)% ⊦ i ≤ j ↔ ¬ j < i; (ss)% ⊦ i < j → ¬ j < i; (s) Used terms: j, i + j0, i1 * j